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An elementary proof of the identity $\cot θ= \frac{1}θ\sum _{k=1}^{\infty} \frac{2 θ}{θ^2 - k^2 π^2}$. (English)
Int. J. Math. Educ. Sci. Technol. 43, No. 8, 1085-1092 (2012).
Summary: This article gives an elementary proof of the famous identity $$\cot θ= \dfrac{1}θ \sum _{k=1}^{\infty} \dfrac{2 θ}{θ^2 - k^2 π^2}, \quad θ\in \Bbb{R}\setminus π\Bbb{Z}.$$ Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Euler’s and Neville’s identities, as well as the identity $ζ(2) := \sum _{k=1}^{\infty}\frac{1}{k^2}= \frac{π^2}{6}$.
Classification: I30 I50